Fast Growing Hierarchy Calculator [upd] ❲Trusted ◎❳

To understand how a calculator processes these levels, we can look at how standard arithmetic operations emerge from the lowest levels of the hierarchy. Level 0: Successor Behavior: Simple counting. Level 1: Multiplication-like Growth Formula: Evaluation: . This yields Behavior: Linear growth. Level 2: Exponential Growth Formula: Evaluation: Doubling a number times yields Behavior: Exponential growth. Level 3: Power Towers (Tetration) Formula:

To understand what a calculator does, you must see how quickly the levels escape ordinary human comprehension. The Low Levels (Finite Ordinals) : Linear growth (Addition). : Linear growth (Multiplication). : Exponential growth. is roughly equal to 64.

101010010 raised to the exponent 10 to the 100th power end-exponent

However, there is a critical nuance:

Instead, an FGH calculator does one of three things: fast growing hierarchy calculator

used to classify the growth rates of extremely large numbers and computable functions. Because these functions grow so rapidly that they quickly exceed physical limits (like the number of atoms in the universe), specialized online calculators are used to explore their values and expansions. Online FGH Calculators

If $\alpha$ is a limit ordinal (like $\omega$ or $\omega \times 2$), we use fundamental sequences. $$f_\alpha(n) = f_\alpha[n](n)$$ Translation for the calculator: Find the $n$-th element in the fundamental sequence of $\alpha$ and evaluate that function.

For those who want to get their hands dirty with code, GitHub hosts Python implementations. The repository mshoosterman/fast-growing-hierarchy is an explicit attempt to implement the Wainer hierarchy. Another repository, JacobDreiling/googology , is a goldmine that includes FGH strengths for many functions, providing a practical way to compare their growth rates.

: For the smallest index, the function is just simple addition. f0(n)=n+1f sub 0 of n equals n plus 1 To understand how a calculator processes these levels,

An FGH calculator is, in a sense, a partial time machine. It lets you skip past the puny exponentials, past the Knuth arrows, past Conway chains, past the busy beaver of low-level recursion, and stare directly at the boundary where computation itself begins to falter.

return "Unknown Ordinal"

Here’s a concept for a , designed for both education and experimentation with large numbers and ordinals.

The hierarchy is defined by three primary rules that govern how functions evolve from basic operations into astronomically large numbers: . This is the successor function. Successor Step . The function at level -th iteration of the function at level applied to Limit Step is a limit ordinal. This process, known as diagonalization , uses the -th term of a fixed fundamental sequence assigned to 2. Common Levels and Growth Rates As the index This yields Behavior: Linear growth

Set-theoretic large number that surpasses the standard Fast-Growing Hierarchy entirely. Architecture of an FGH Calculator

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), and the Bachmann-Howard ordinal. These levels track functions like the Tree function and Subcubic Graph numbers. How to Use an FGH Calculator

, and up to the and the Feferman-Schütte ordinal ( Γ0cap gamma sub 0 ) , pushing the boundaries of what can be logically defined. Anatomy of a Fast-Growing Hierarchy Calculator